Trigonometric identities are relationships between trigonometric ratios that define them in terms of one another. They can be used to help solve problems that involve trigonometric functions.
The reciprocal function to the sine is the cosecant, to the cosine is the secant, and to the tangent is the cotangent. In math language, the csc x = 1/sin x; sec x = 1/cos x, and cot x = 1/tan x. If tan x =sin x/cos x, then cot x also equals cos x/sin x. The reciprocal identities are helpful because all trigonometric ratios in a problem can be rewritten in terms of sines and cosines, which are often much easier to use.
The Pythagorean identities are sin2x + cos2=1; tan2x +1 = sec2x; and 1 + cot2x = csc2x. Using the Pythagorean identities, one expression can be solved in terms of another by using substitution. The important thing to remember is that the ratios are squared ratios, similar to using the Pythagorean Theorem with right triangles.
Using Known Information
One way to solve a trigonometric expression is to make sure that any rules of algebra are followed. For example, fractions can be put into common denominators and combined, using like denominators. Suppose that the fractions to be added are sin θ/cos θ +cos θ/ (1 + sin θ). In order to add the fractions, they must have common denominators. Therefore, (sin θ [1 +sin θ] +cos2θ)/ (cos θ (1 + sin θ) can be simplified to 1/cos θ, which is sec θ.
Simplifying Trigonometric Expressions
Another key to simplifying trigonometric expressions is to write them in terms of sines and cosines, because they are often easier to use. Tables are written with values of sines and cosines. Also, many of the common formulas, such as the area of a triangle, express values using the sine of an angle, and the Law of Sines and the Law of Cosines can be used to solve values for the triangle. Applications are much easier to use when the trigonometric expression is simplified.
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|sine(q) = opp/hyp||cosecant(q) = hyp/opp|
|cosine(q) = adj/hyp||secant(q) = hyp/adj|
|tangent(q) = opp/adj||cotangent(q) = adj/opp|
The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
It is often simpler to memorize the the trig functions in terms of only sine and cosine:
|sine(q) = opp/hyp||csc(q) = 1/sin(q)|
|cos(q) = adj/hyp||sec(q) = 1/COs(q)|
|tan(q) = sin(q)/cos(q)||cot(q) = 1/tan(q)|
|arcsine(opp/hyp) = q||arccosecant(hyp/opp) = q|
|arccosine(adj/hyp) = q||arcsecant(hyp/adj) = q|
|arctangent(opp/adj) = q||arccotangent(adj/opp) = q|
The functions are usually abbreviated:
According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 arccsc-1, arcsec-1, and arccot-1. Beware, though, there is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
See also: overview.